WSP10809

• File Number: 8273.500
• Description: Colorado River Basin Salinity Control - Federal Agency Reports - EPA
• Basin: Colorado Mainstem
• Water Division: 5
• Date: 4/1/1978
• Title: Integrating Desalination and Agricultural Salinity Control Alternatives
• Water Supply Pro - Doc Type: Report/Study

where the gradient, (V f), is called the Jacobian Matrix, J,
~ and the matrix (Vdf) c~n be relabeled as C. Employing these
00 variables in Eq.7 and rearranging terms: -
l-"
'0 ~a~ = -~a~ +a~ ........................................(8)

The vector as can be solved for if the Jacobian matrix is always
taken non-singular:

_1 -1
ge.l~).J~l~. ~aS!. + J a1.................................. (9)

The elimination of the states is now possible by sub-
stitution of Eq. 9 into Eq. 6. After rearranging terms, the
final unconstrained equation is developed:

ay = [VdY - (Vsy)~_l~] ad + (Vsy)[lai ...............(10)

Kuhn-Tucker Conditions --

By definition of the total differential, another expression
can be written in terms of the variables indicated in Eq. 10.
If the elimination of the state differentials were accomplished
then the total differential of y would be written:

ay = ~ a~ + ~~ act> ..................................(11)

in which 8y/8d and 8y/8cp are called "constrained derivatives."
The deviation-in notation is made to distinguish the ay/ax,
which is a partial derivative viewing all variables as inde-
pendent, from 8y/8d which is a partial derivative considering T
of the variables as functions of the remaining N variables.
By comparing Eqs. lO and II it can be seen that,

~ _1
ad = 'VdY - ('i.Jsy)~ C .................................(12)

and,

~ _ -1
oct> - ('Vsy)~ .........................................(13}

The solution of Eqs. 12 and 13 when equated to zero yield a
stationary point when the decision variables are free, or in
other words, allowed to assume any positive or negative value.
In most instances, decision variables are not free, but subject
to non-negativity conditions. Stationary points may be local
or global minimums, maximums, or inflection points. The evalu-
ation of stationary points in these cases will depend on
criteria reported by Kuhn and Tucker (1951) which provide neces..
sary and sufficient conditions for a minimum. In the problem
solution at the feasible point u~der examination, a minimum
exists if the following conditions are met:

12